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Tokyo Probability Seminar
Seminar information archive ～09/06｜Next seminar｜Future seminars 09/07～
| Date, time & place | Monday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Makiko Sasada, Shuta Nakajima (Keio Univ.), Masato Hoshino (Science Tokyo), Masahisa Ebina (Science Tokyo) |
Future seminars
2026/09/10
14:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Lectures start earlier.
Sung-Soo Byun (Seoul National University) 14:00-15:30
Inhomogeneous Geometric Last Passage Percolation and the q-deformed Jacobi Unitary Ensemble
GIG random matrices and a Yang-Baxter extension of the Matsumoto-Yor property
Lectures start earlier.
Sung-Soo Byun (Seoul National University) 14:00-15:30
Inhomogeneous Geometric Last Passage Percolation and the q-deformed Jacobi Unitary Ensemble
[ Abstract ]
Since the pioneering work of Johansson, geometric last passage percolation (LPP) has become one of the central models in integrable probability, with many remarkable properties established over the past two decades. Most existing results, particularly those admitting explicit formulae, however, concern homogeneous environments. In this talk, I will discuss geometric LPP in a particular inhomogeneous environment. Our approach is based on a duality with a q-deformed analogue of the classical Jacobi unitary ensemble. Exploiting recent developments in the asymptotic spectral analysis of this random matrix model, I will present recent results on the law of large numbers, an explicit characterisation of the shape function, and the fluctuation behaviour of the last passage time.
Jacek Wesolowski (Warsaw University of Technology) 16:00-17:30Since the pioneering work of Johansson, geometric last passage percolation (LPP) has become one of the central models in integrable probability, with many remarkable properties established over the past two decades. Most existing results, particularly those admitting explicit formulae, however, concern homogeneous environments. In this talk, I will discuss geometric LPP in a particular inhomogeneous environment. Our approach is based on a duality with a q-deformed analogue of the classical Jacobi unitary ensemble. Exploiting recent developments in the asymptotic spectral analysis of this random matrix model, I will present recent results on the law of large numbers, an explicit characterisation of the shape function, and the fluctuation behaviour of the last passage time.
GIG random matrices and a Yang-Baxter extension of the Matsumoto-Yor property
[ Abstract ]
A map is independence preserving (IP) if there exists a product probability measure which is transferred by this map into to a product probability measures. Many classical examples of such maps and related probability laws are known probably the best known is the map (x,y)\mapsto (x+y, x-y) is an IP map for product of normal laws. Sasada and Uozumi (2024) identified IP a new family of parametric Yang-Baxter maps on positive quadrant together with related probability laws. In particular, the map H_{III,B}^{(\alpha,\beta)}, of this family was connected to the IP property of the GIG distributions. This IP property appears also naturally in probabilistic integrable models of discrete Korteweg de Vries type, as observed by Croydon and Sasada (2020). In the case of (\alpha,\beta)=(1,0), remarkably, this IP property reduces to the classical Matsumoto-Yor property rooted in the conditional structure of functionals of exponential Brownian motion.
We will propose an extension of H_{III,B}^{(\alpha,\beta)} to a Yang-Baxter map on the cone of symmetric positive definite matrices of a fixed dimension. We will show that this extended map preserves independence of GIG random matrices. We will present a proof that the matrix GIG distribution is characterized by the IP property of this map.
A map is independence preserving (IP) if there exists a product probability measure which is transferred by this map into to a product probability measures. Many classical examples of such maps and related probability laws are known probably the best known is the map (x,y)\mapsto (x+y, x-y) is an IP map for product of normal laws. Sasada and Uozumi (2024) identified IP a new family of parametric Yang-Baxter maps on positive quadrant together with related probability laws. In particular, the map H_{III,B}^{(\alpha,\beta)}, of this family was connected to the IP property of the GIG distributions. This IP property appears also naturally in probabilistic integrable models of discrete Korteweg de Vries type, as observed by Croydon and Sasada (2020). In the case of (\alpha,\beta)=(1,0), remarkably, this IP property reduces to the classical Matsumoto-Yor property rooted in the conditional structure of functionals of exponential Brownian motion.
We will propose an extension of H_{III,B}^{(\alpha,\beta)} to a Yang-Baxter map on the cone of symmetric positive definite matrices of a fixed dimension. We will show that this extended map preserves independence of GIG random matrices. We will present a proof that the matrix GIG distribution is characterized by the IP property of this map.
